Experimental Identification of Friction and Dynamic Coupling in a Dual Actuator Testbed

Transcription

1 Experimental Identification of Friction and Dynamic Coupling in a Dual Actuator Testbed Dinesh Rabindran 1 and Delbert Tesar 2 1 Ph.D. Candidate, 2 Director and Carol Cockrell Curran Chair Robotics Research Group, Mechanical Engineering Department The University of Texas at Austin, Austin, TX dineshr@mail.utexas.edu Abstract In this paper our objectives were to present the methodology and results for the experimental identification of two performance-limiting phenomena in a differential-based dual-input actuator testbed: (i) velocity- and position-dependent friction and (ii) dynamic coupling between the dual inputs. The contributions of this paper are as follows: (i) experimental methodology to characterize the dynamic coupling in the dualinput actuator and (ii) methodology and data analysis for friction identification in a dual input actuator. We have leveraged and replicated experimental procedures on friction identification reported in the literature. The Stribeck-effect and viscous damping effects were determined with an average repeatability of 3σ = 5.6% using constant velocity experiments. Position dependent friction was determined using spatial FFT analysis of torque data with a variance of 3σ = 0.56%. Dynamic coupling was determined using an experimental set up where the torque disturbance on one branch of the dual-actuator is measured during the periodic motion of the other. Crosscorrelation analysis determined that, in our testbed, the coupling was well correlated (r = and 3σ = 3.6%) with velocity of motion. The implication of our results is that the methodology and data analysis techniques outlined in this paper provide a first step towards the experimental certification of dual-input actuators. Index Terms Friction Identification, Dynamic Coupling, Dual Actuator, Experimental Methodology I. INTRODUCTION Differential-based dual-input actuators have been proposed in the past [1], [2] and the performance-limiting effects of friction in dual actuators have been investigated [1]. The Parallel Force/Velocity Actuator (PFVA) was proposed [3] and experimentally demonstrated by Rabindran and Tesar [4]. The PFVA is a differential-based dual input single output actuator. In this paper, our objectives are to present the methodology and results for the experimental identification of two performancelimiting phenomena in a PFVA prototype and testbed: (i) velocity- and position-dependent friction, and (ii) dynamic coupling between the dual inputs. We have leveraged and replicated experimental procedures reported in the literature on friction identification [5], [6]. Our work is reported according to the guidelines laid down in [7]. The contributions of this work are as follows: (i) experimental methodology to characterize the dynamic coupling in the dual-input actuator and (ii) methodology and data analysis for friction identification in a dual input actuator. Fig. 1. PFVA Testbed Layout and Components II. PFVA TESTBED DESCRIPTION The differential-based PFVA consists of two inputs: Force Actuator (FA) and Velocity Actuator (VA). For details of the concept, see [8]. The PFVA testbed is shown in Fig. 1. The principal components of this system are (i) the FA and VA motors, (ii) a 2-DOF differential gear train [9] that mixes inputs from these motors, and (iii) An in-line torque sensor in the FA branch. The FA and VA have velocity ratios of g f = and g v = to the output, respectively. The ratio of velocity ratios of the two inputs will be called the Relative Scale Factor (RSF) [10] or ρ: ρ = g f g v (1) A detailed list of components and their specifications has been compiled in Table I. III. EXPERIMENT I: IDENTIFICATION OF FA FRICTION The objective of this experiment was to characterize the Stribeck effect at low-velocities [5], viscous damping at higher velocities, stiction, and position-dependent friction, all in the FA branch of the PFVA because the FA is a near directdrive input and friction is more pronounced in this subsystem. Furthermore, the velocity-dependent friction effects

2 TABLE I LIST OF PFVA TESTBED COMPONENTS Component Manufacturer Specification VA Motor Kollmorgen τ max = 28.9 N-m (Framed) RBE A50 ω max = 281 rpm Enc Counts/Rev = 8192 FA Motor Kollmorgen τ max = 150 N-m (Framed) DH063M ω max = 800 rpm Enc Counts/Rev = 8192 Torque Sensor Honeywell Lebow τ max = 200 N-m NM Accuracy = 0.25% Rise-Time = 2ms 2-DOF Differential Andantex τ max = 150 N-m (Oil Lubricated) SR-20 RSF ρ = 24.3 Torque Sensor Reading Measured Output Link Removed Home Position Indicator FA Motor Controlled at Different Velocities VA Motor Controlled at Zero Velocity Fig. 2. Conditions imposed on the PFVA testbed during friction identification experiment. VA motor was controlled at zero velocity, FA motor was controlled at different velocities ranging from -200 to 200 rpm, and the torque reading from the torque sensor was measured. are different in low- and high-velocity zones. In the highvelocity region, viscous damping is predominant and in the low-velocity region, the Stribeck effect [5] is present. Positiondependent friction [6] arises due to inaccuracies in the assembly of the testbed and the resulting loading on the FA shaft as a function of its angular position. Our goal was to leverage past work in experimental friction identification [5], [6] to systematically lay out an experimental procedure and estimate the above mentioned friction effects magnitudes in the FA branch. The theory behind this procedure is the Stribeck friction model [5]. The difference in methodology between our work and that by Garcia et al. [6] is that they indirectly compute the friction torque by measuring the motor current; however we directly measure the torque using a torque sensor in the FA branch (see Fig. 1). A. Methodology To determine the velocity-dependent friction, the VA motor was controlled at zero velocity (holding position), the FA motor was controlled at different velocities in the range 200 φ f 200 rpm. The lowest non-zero velocity experimented with was 5 rpm. For each FA velocity value, 10 runs were performed and the mean torque measurement was determined. For each run, the torque was sampled for 15 revolutions of the FA shaft. Torque was sampled at 20 Hz and a second-order low-pass forward-backward Butterworth filter [11] with a cut-off frequency of 5 Hz was used. To eliminate the artifacts in the filtered signal due to forwardbackward filtering, the torque data from the first two and the last two periods of the shaft revolution were ignored for averaging. This process also eliminates transients in the torque measurements due to the PID action of the motor controller. For every speed setting, the average and variance of torque measurements from all 10 runs were, respectively, used as estimates of the frictional torque and its repeatability at that speed. Before this experiment, a warm up routine was used where the FA was run for approximately 2 minutes at 200 rpm in both directions to eliminate error due to temperature variation. The significance of warming up is that friction decreases rapidly after a short period (1-2 mins) of activity across the whole range of velocities. This is explained in detail by Armstrong-Hélouvry, 1991 [5] (Chapter 5). The output link was removed for this experiment to eliminate gravity loading due to its weight. The FA velocity values selected for different runs were randomized to eliminate experimental bias. To determine the stiction torque, the current on the FA motor was gradually increased while monitoring for the movement of the FA shaft. The torque measurement at the instant when the FA shaft started moving (break-away torque) was used as an estimate of stiction. This stiction experiment was conducted for 8 runs each for both positive- and negative- torque regions to determine the variance (or repeatability) of the stiction estimate. To measure position-dependent friction, it was important to choose a reference to count rotations of the FA shaft to then characterize the friction torque as a function of angular shaft position. Therefore, a home position for the FA was arbitrarily chosen as a reference and marked on the testbed as shown in Fig. 2. The procedure followed for this experiment was similar to the one for measuring velocity dependent friction. The VA was controlled at zero velocity. The FA was controlled at different velocities chosen from the set {±5, ±10, ±25} rpm and was controlled to repeatably move for exactly 12 revolutions during every experiment. Torque data was sampled at 20 Hz and a low-pass second-order butterworth forwardbackward filter [11] with a cut-off frequency of 5 Hz was used for data analysis. To verify that the friction torque is dependent on the angular position, the spatial frequency spectrum, or Fast Fourier Transform (FFT), of the torque data was plotted to compare the frequency content in the torque data and the angular frequency of rotation of the FA shaft. The unit used for spatial frequency was cycles/rev (as opposed to cycles/second for temporal frequency). This change of units allows us to focus on the torque oscillations as a function of angular position (in terms of revolutions). The procedure described above was laid down in a monograph by Armstrong-Héouvry [5] and later used by Garcia et al. [6] to determine the positiondependent friction in the joint of a legged robot.

3 Stochastic Nature of Friction at Low Velocities Fig. 3. Experimental results for velocity-dependence of friction in the FA branch. Notice the Stribeck effect in the low velocity region where average friction torque decreases with velocity. After the critical velocity, friction increases linearly (viscous damping effect). Error bars show 3σ intervals. Fig. 4. Results from repeatability analysis of our velocity-dependence experiments. The bars in this figure show the value of 3σ in percentage for the experiment at every speed setting. Based on this bar chart, the average 3σ for all experiments was 5.6% which indicates a relatively high repeatability. B. Data Analysis and Results The results from our velocity-dependent friction estimation experiments are shown in Fig. 3 (friction torque vs. speed range) and Fig. 4 (repeatability of velocity-dependent friction experiment). From our break-away experiments, the positive and negative stiction torque measurements were and N-m, respectively, with a repeatability of approximately 3σ = 7% (based on the variance across the 8 runs performed). The error bar in Fig. 4 shows the repeatability of torque measurement at a speed (based on the 3σ computation across the 10 runs performed). There are three noteworthy observations from the first quadrant in Fig. 3: The segment from approximately to rad/s shows a linear viscous damping trend with correlation coefficient of The viscous damping coefficient in this region is N-m/rad/s. The Stribeck effect was observed with an estimated critical velocity of rad/s. After break-away, the friction torque decreases from its stiction value (1.008±7%) to approximately ±4%. This rate of decrease is approximately N-m/rad/s. Correspondingly, in the third quadrant the following observations were made: The segment from approximately to rad/s shows a linear viscous damping trend with correlation coefficient of The viscous damping coefficient in this region is N-m/rad/s. The Stribeck effect was observed with an estimated critical velocity of rad/s. After break-away the friction torque decreases from its stiction value ( ± 7%) to approximately ± 9%. This rate of decrease is approximately N-m/rad/s. We also performed a repeatability analysis for our experiment at various speeds (see Fig. 4). The variance in the low velocity region (< 5 rad/s in both positive and negative directions) was on an average approximately 2 times lower than that for higher velocities ( 5 rad/s). This relatively lower repeatability at these lower velocities could be explained as follows: (i) friction has systematic and stochastic components [5] and at low velocities the stochastic behavior might be predominant, (ii) the torques observed in these experiments are approximately 0.2% of the torque range of the sensor and, therefore, the sensor readings are relatively less precise, and (iii) the variance was calculated across a small sample (10 runs) and our expectation is that the repeatability would improve with a larger sample. Our main focus here was to lay out a systematic method to characterize the velocity-dependent friction in the FA branch for very low velocities. We have only experimented with speeds as low as 25% of the rated speed of the FA motor. This is because the FA is expected to be in a low velocity zone for a significant portion of its operation (see Chapter 6 in [4]). As mentioned earlier, the FFT analysis (see Fig. 5) was performed on the filtered torque data expressed as a function of FA shaft position. The FFT was plotted using spatial frequency units (cycles/rev). The advantage of using these units is twofold: The frequency of torque oscillation as a function of shaft revolution (ν) allows us to compare energy content in the torque data at various multiples of a rotation. A peak in the FFT magnitude at ν = 1 cycles/rev (fairly repeatably at various speed settings) suggests that there is an oscillation of friction torque during every rotation of the FA shaft. Typically, if a peak is observed at a frequency ν other than ν = 1 (again, for various speed settings) then the ratio r ν ν = ν ν is indicative of torque oscillations caused due to another component which is rotating at the rate of r ν ν revolutions when the FA shaft rotates one revolution.

4 TABLE II SUMMARY OF IDENTIFIED FRICTION PARAMETERS Positive Negative Stiction Torque (N-m) Viscous Damping (N-m/rad/s) Viscous Damping Correlation Critical Velocity (rad/s) Position-Dependent Torque ν = 1, 2 ν = 1, 2 Oscillation Frequency (cyc/rev) Torque Sensor Reading Measured FA Motor Controlled at Zero Velocity In other words, it is possible to back out gear ratios in a system. For instance, in the above mentioned case, r ν ν = 2 could possibly be a gear ratio in the system. See [6] for details. In our FFT results shown in Fig. 5, we observed a peak frequency at ν 1 = 1 cycles/rev (first dashed blue line) with a 3σ limit of 0.56%. This shows a very high repeatability of torque oscillation at the rate of rotation of the shaft. A second peak frequency of ν 2 = 2 cycles/rev (second dashed blue line) was observed with a 3σ limit of 0.54%. This suggests that the rotation of a component at approximately 2 times the revolution of the FA shaft is causing this frequency of oscillation. Although the detailed design of the gear train, and thus the intermediate gear ratios, are not available to us, we suspect that this oscillation could be generated by a component in the differential gear train. In addition to the FFT plots, we have also included the plots of low-pass filtered torque data w.r.t. FA shaft position (see Fig. 6) for the various speed settings experimented with and for 10 revolutions of the FA shaft. Torque oscillations can be caused due to many phenomena such as the stress in the shaft-coupler and deflection in the bearing. Even gravity loading due to unbalanced masses could contribute to cyclic loading. In this section we have focussed on laying down a systematic procedure to experimentally characterize some of these position-dependent phenomena in a potential, and more refined, second prototype of the PFVA. The friction parameters identified using Experiment I have been summarized in Table II. IV. EXPERIMENT II: IDENTIFICATION OF DYNAMIC COUPLING There were two important research questions that we intended to answer with this experiment: (i) how much coupling torque (mutual disturbance) is felt by the inputs of a differentially summed dual actuator, and (ii) which motion parameter (position, velocity, or acceleration) does this coupling torque correlate with? Knowledge of this coupling torque is essential to designing a control scheme for real-time operation of the PFVA. However, to the best knowledge of the authors, no experimental studies have been reported in the literature to identify dynamic coupling between dual inputs although its performance-limiting effects are evident - mutual disturbances during operation. Therefore, we believe that the following experimental methodology and data analysis technique to answer the above questions is a contribution to the experimental Output Link Removed VA Motor Controlled to Track Simple Harmonic Motion Fig. 7. Conditions imposed on the PFVA testbed during dynamic coupling identification experiment. FA motor was controlled at zero velocity, VA motor was controlled to track simple harmonic motion, and the torque reading from the torque sensor was measured. literature in robotic actuators. A. Methodology The theory behind this experiment can be written in the form of two coupled differential equations: I vv φv + I vf φf + v τ F + v τ G + g t 1 ρ + 1 τ o = τ Mv I fv φv + I ff φf + f τ F + f τ G + ρ ρ + 1 τ o = τ Mf (2a) (2b) where i τ G (φ v, φ f ) and i τ F (φ v, φ f, φ v, φ f ) are, respectively, the gravity and friction torques for input i {v, f}, g t is the velocity ratio of the differential s casing w.r.t the VA motor shaft, I vv is the principal inertia on the VA side, I ff is the principal inertia on the FA side, I vf = I fv is the coupling inertia between the VA and FA, and φ v and φ f are, respectively, the VA and FA shaft positions. The conditions imposed on the PFVA testbed were as follows: The VA motor was commanded to perform Simple Harmonic Motion (SHM) φv = Ω sin(2πν + ϕ) with frequency ν and amplitude Ω from the sets { } Hz and { } rpm, respectively. Without loss of generality, it was assumed that ϕ = 0. Considering all combinations of frequency and amplitude values from the above sets, a total of 9 readings were taken which were randomized to eliminate experimental bias. The FA motor was controlled using a stiff velocity control loop to maintain position (i.e. φ f = φ f = φ f = 0). Therefore the FA motor controller provides just enough torque to compensate the disturbance introduced by the VA s SHM. Under the above conditions of the VA and FA, the coupling torque between them is measured by the torque sensor (sampled at 20 Hz) in the FA branch because the FA shaft is controlled to be at rest. This measured torque

5 Fig. 5. Results from FFT analysis of friction torque data as a function of FA shaft position during the experimental determination of position-dependence of friction. Note that the frequency units are cycles/rev to directly determine torque oscillation period as a multiple of FA shaft revolution. Fig. 6. Torque data plotted with respect to FA shaft position for the six test speeds and for 10 revolutions. A low-pass second-order Butterworth filter was used with a cut-off frequency of 5 Hz.

6 TABLE III CROSS-CORRELATION ANALYSIS SUMMARY Torque-Position Torque-Velocity Torque-Acceleration Correlation Variance (%) estimates a combination of (i) the inertial coupling torque on the FA motor due to the acceleration of the VA motor shaft, (ii) the coupling viscous friction torque on the FA motor due to the velocity of the VA, and, possibly, (iii) the disturbance on the FA dependent on the position of the VA. The output link was removed to eliminate gravity loads. To determine the motion parameter (i.e. position, velocity, or acceleration) which predominantly contributes to the coupling torque, a cross-correlation analysis was performed between three pairs of measured signals: (i) torque vs. VA acceleration, (ii) torque vs. VA velocity, and (iii) torque vs. VA position. The cross-correlation between two signals x(t) and y(t) is defined as [12] r xy (τ) = which, for discrete-time signals, reduces to r xy (m) = 1 N N 1 n=0 x(t)y(t τ)dt (3) x(n)y(n m) (4) The order of the subscripts in Eq.(3) indicates that x is unshifted while y is shifted. The parameter m is the time-shift or lag and the maximum lag introduced in our experiment for the frequencies 0.25, 0.5, and 1 Hz were 300, 200, and 100, respectively. Signal noise was filtered using a secondorder forward-backward Butterworth filter. It is important to do both forward and backward filtering to eliminate the lag introduced by the filter. This lag would bias our cross-correlation results. Note that, however, forward-backward filtering can be performed only in off-line situations such as the analysis for this experiment. Another artifact introduced by filtering is the transient at the beginning. Therefore, the experiment was run for exactly 12 oscillations for every amplitude and frequency combination, and the first and last time-periods were ignored during cross-correlation. B. Data Analysis and Results The time- and frequency-domain plots of the disturbance torque τ vf felt by the FA are shown in Fig. 8(a) and Fig. 8(b), respectively. The plot of the VA SHM is shown in Fig. 8(c). Notice the relatively high peak in the FFT magnitude of the disturbance torque at 0.25 Hz, the SHM frequency of the VA. This indicates that the VA s motion might have an influence on the disturbance torque felt by the FA motor. Since all three motion parameters (position, velocity, and acceleration) of the VA have the same frequency in this experiment, it is necessary to do a cross-correlation analysis of τ vf with each of φ v, φv, and φ v. (a) (b) (c) Fig. 8. Torque sensor measurements for Experiment II when VA velocity was cycled at 0.25 Hz with an amplitude of 5.23 rad/s. (a) Time-domain torque data, (b) FFT of torque data, and (c) VA velocity data in time-domain.

7 Fig. 9. Cross-correlation results for coupling torque data w.r.t. VA position, velocity, and acceleration for various frequencies (0.25, 0.5, and 1 Hz) and 5.23 rad/s amplitude. As an example result, the cross-correlation function magnitudes for various lag values are shown in Fig. 9. The first, second, and third rows in this figure correspond to correlation of τ vf with position, velocity, and acceleration of the VA, respectively. For instance, in the bottom right corner of this figure is shown the correlation magnitude (r ταv ) and cross-correlation results for 5.23 rad/s and 0.25 Hz between τ vf and the VA acceleration for this setting. These results suggest that, for our experiment, the disturbance torque is strongly correlated (for example, r τωv = for 1 Hz) with the velocity of the VA and, at the same time, weakly correlated with the acceleration (r ταv = 0.044) and position signals (r τφv = 0.145). These results were also produced for the other speed settings. The cross-correlation data was tabulated (see Table III) to examine the repeatability of this result. It was observed that the mean correlations (over the 9 settings for torque vs. position, velocity, and accelerations were, respectively, (±60%), (±1.22%), and (±70%). There was poor repeatability in the position and acceleration correlations possibly due to a small sample of 9 readings. On the contrary, the torque to velocity correlation was very strong and repeatable in spite of the small sample size. The physical meaning of this result can be investigated by partitioning τ vf : τ vf = I vf φv + τ vf ( φ v ) + τ vf (φ v ) (5) Now, the first term on the right hand side of Eq.(5) is dependent on the inertial coupling between the two inputs which in turn is a function of the RSF ρ as shown in Fig. 10 [13]. In our testbed, ρ = 24.3 which according to the model shown in Fig. 10 corresponds to µ = This results in a very low disturbance torque component due to inertias. We hypothesize that this could be a prominent reason for the weak correlation between τ vf and VA acceleration. The weak correlation with position can be explained by the fact that there is neither gravity loading nor positiondependent friction from the FA branch. The strong correlation of the disturbance torque with velocity is probably because the FA motor s PID velocity-controller is reacting primarily to velocity-disturbances acting on the FA shaft more than any other kind of influence. Therefore the FA motor s active torque, and thus, the torque measured by the torque sensor follow the trend of the VA velocity profile. This hypothesis can be easily tested by monitoring the current on the FA motor. Future experiments might benefit from the presence of an accelerometer in the FA branch and another torque sensor in the VA branch. In summary, the objectives of Experiment II were to: confirm the presence of a dynamic coupling phenomenon.

8 RSF = 24.3 for PFVA Prototype in Experiment II Fig. 10. Variation of the dynamic coupling factor between the FA and VA and its derivative as a function of the RSF ρ. Note that as lim µ = 0 and ρ lim dµ/d ρ = 0. This figure is adapted from [13]. ρ This is a fairly intuitive behavior in a dual velocitysumming mechanism; however it was important for us to characterize it. design an experiment to measure this phenomenon. As explained earlier, due to the conditions imposed on the FA motor there is a good chance (98.77%) that the measured disturbance torque follows the same trend as the VA velocity due to the PID action of the FA motor. However, the methodology laid out in this section is a first step towards a more refined characterization of dynamic coupling. compare the relative contributions of VA position, velocity, and acceleration to the torque disturbance on the FA branch. According to the results presented in Table III, the VA velocity is approximately 2 orders of magnitude better correlated with the disturbance torque than the position or acceleration. V. SUMMARY AND DISCUSSION Our emphasis in this paper was on experimentally identifying two performance-limiting phenomena in a differentiallysummed dual-actuator (or PFVA): (i) friction in the directdrive (or FA) branch and (ii) dynamic coupling torque between the two inputs of the PFVA. For friction experiments we replicated the experimental methodology laid down by past work [6]. Additionally, we proposed and demonstrated a new experimental method to identify dynamic coupling between the dual inputs of a differentially summed actuator and its correlation with motion parameters. Friction phenomena, i.e. position- and velocity-dependent friction, and stiction, were identified with good repeatability (3σ = 5.6%). A good correlation ( ) was observed between the dynamic coupling torque experienced by the FA and the velocity of the VA. The repeatability of this observation was 3σ = 3.6%. We believe that it is important to experimentally certify robotic actuators to improve the reliability of system operation. Some work has been done inside [14] and outside [15] of our research group in this area for single-input geared actuators. In this paper we present preliminary work in experimentally identifying performance-limiting effects of differential-based dual-input actuators. ACKNOWLEDGMENT This project was supported by the DOE grant DE-FG52-06NA2559. The assistance provided by P. Donner and M. Pryor for the set up of the PFVA testbed is acknowledged. P. Donner s assistance with break-away experiments is acknowledged. REFERENCES [1] J. Ontanon-Ruiz, P. McAree, and R. 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